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Chapter 5: Problem 51

There are 100 employees at Kiddie Carts International. Fifty-seven of the employees are hourly workers, 40 are supervisors, 2 are secretaries, and the remaining employee is the president. Suppose an employee is selected: a. What is the probability the selected employee is an hourly worker? b. What is the probability the selected employee is either an hourly worker or a supervisor? c. Refer to part (b). Are these events mutually exclusive? d. What is the probability the selected employee is neither an hourly worker nor a supervisor?

Short Answer

Expert verified
a. 0.57 b. 0.97 c. Yes, they are mutually exclusive. d. 0.03

Step by step solution

01

Total Employees

Identify and confirm the total number of employees at Kiddie Carts International. The total number of employees given is 100.
02

Probability of Selecting an Hourly Worker

Calculate the probability of randomly selecting an hourly worker.There are 57 hourly workers.The probability is calculated as: \[ P(\text{Hourly Worker}) = \frac{\text{Number of Hourly Workers}}{\text{Total Employees}} = \frac{57}{100} = 0.57 \]
03

Combined Probability of Hourly Worker or Supervisor

Calculate the probability of selecting an employee who is either an hourly worker or a supervisor.There are 57 hourly workers and 40 supervisors. Since they do not overlap, we sum the probabilities:\[ P(\text{Hourly Worker or Supervisor}) = \frac{57 + 40}{100} = \frac{97}{100} = 0.97 \]
04

Check If Events Are Mutually Exclusive

Determine if the events 'hourly worker' and 'supervisor' are mutually exclusive. Mutually exclusive events cannot occur simultaneously. Since an employee cannot be both an hourly worker and a supervisor, these two events are mutually exclusive.
05

Probability of Neither Hourly Worker Nor Supervisor

Calculate the probability that the selected employee is neither an hourly worker nor a supervisor.Since there are 97 employees who are either hourly workers or supervisors, there are 3 employees who are neither.The probability is:\[ P(\text{Neither Hourly Worker Nor Supervisor}) = \frac{3}{100} = 0.03 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mutually Exclusive Events
Mutually exclusive events are events that cannot happen at the same time. In this context, if we consider two events - selecting an hourly worker or selecting a supervisor - these are mutually exclusive. This is because an employee cannot be both an hourly worker and a supervisor simultaneously. Understanding mutually exclusive events is crucial in probability as it helps determine how probabilities are combined. For mutually exclusive events, you can simply add their individual probabilities to find the combined probability:
\[ P(A \text{ or } B) = P(A) + P(B) \]
Here, selecting an hourly worker or a supervisor does not overlap, hence confirming their mutual exclusivity.
Probability Calculation
Calculating probability involves determining the likelihood of an event occurring based on the possible outcomes in a sample space. The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. This forms the probability formula:
\[ P( ext{Event}) = \frac{ ext{Number of Favorable Outcomes}}{ ext{Total Number of Outcomes}} \]
Applying this to finding the probability of selecting an hourly worker, we divide the number of hourly workers (57) by the total number of employees (100). So, the probability is 0.57, or 57%. This step-by-step approach helps in clearly delineating the chance of a specific outcome.
Event Intersection
The concept of event intersection pertains to events that can occur simultaneously, unlike mutually exclusive events. When we speak about intersection in probability, it means finding outcomes that both events share. Mathematically, we use the intersection symbol \(\cap\), and the probability of intersection is given by:
\[ P(A \cap B) \]
In the given exercise, events were evaluated as mutually exclusive, meaning the intersection is zero since employees cannot be both hourly workers and supervisors at the same time. Understanding this distinction aids in grasping why the probability of the union of mutually exclusive events is the sum of their individual probabilities.
Sample Space in Probability
In probability, the sample space is the set of all possible outcomes of a probability experiment. For the exercise above, the sample space is the total number of employees (100), representing all the possible employees that can be selected. This forms the basis for calculating probabilities.
The sample space must be well-defined to ensure accuracy in calculating probabilities, as seen when we find the probability of an employee being neither an hourly worker nor a supervisor. Knowing the total, we subtract the workers who meet those conditions from the sample space, yielding a probability of 0.03 for selecting an employee who is neither.

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